Height series of Hα spectra in solar limb spicules obtained with the 53 cm coronagraph of the Abastumani Astrophysical Observatory are analyzed. Each height series covered 8 different heights beginning at 3800 km above the photosphere. The spatial difference between neighboring heights was 1'', consequently ~3800-8700 km distance above the photosphere has been covered. The total time duration of each height series was 7 s. We found that nearly 20% of measured height series show a periodic spatial distribution of Doppler velocities. We suggest that this spatial periodicity in Doppler velocity is caused by propagating kink waves in spicules. The wave length is found to be ~3500 km. However the wave length tends to be ~1000 km at the photosphere due to the height variation of the kink speed. This probably indicates to a granular origin for the waves. The period of waves is estimated to be in the range of 35-70 s. These waves may carry photospheric energy into the corona, therefore can be of importance in coronal heating.